The factorial can be neatly expressed in Eindhoven notation (a lot more ASCII-friendly than the pi notation): `n`! = (* : (`i` ∈ **Z**) ∧ (0 < `i` ≤ `n`) : i).

Among other things, the factorial of `n` is the number of different ways `n` items can be ordered. For example, take 3 items, call them `a`, `b` and `c`. These can be arranged in 6 different ways, `a``b``c`, `a``c``b`, `b``a``c`, `b``c``a`, `c``a``b`, or `c``b``a`; 3! = 6. 20 items can be ordered in 20! different ways, etc..

Logically, this makes perfect sense. When choosing the first item, there are `n` to choose from. When choosing the second, a totally independent choice, there are (`n` - 1) items to choose from. Just *which* (`n` - 1) items *in particular* depend on the first choice, but the **number** of items left for the second choice is always the same. Because the two choices are independent, the numbers are multiplied. For the third choice, there are (`n` - 2) items to choose from, etc., and for the last choice there is only 1 item to choose from. So the total number of ways to order them are (`n` * (`n` - 1) * (`n` - 2) … * 1) = `n`!.